We consider only finite undirected graphs without loops or multiple edges.
Notation and terminology not defined here follow those in [1]}.
Let \(G = (V(G), \, E(G))\)  be a graph. A matching \(M\)  in \(G\)  is a set of pairwise non-adjacent edges.
A maximum matching is a matching that contains the largest possible number of edges.
The matching number, denoted \(m(G)\) , of a graph \(G\)  is the size of a maximum matching.
For a vertex \(u\)  and a vertex subset \(U\)  in \(G\) , we use \(N_U(u)\)  to denote all the neighbors of \(u\)  in \(U\) .
We use \(G_1 \vee G_2\)  to denote the the join of two disjoint graphs \(G_1\)  and \(G_2\) .
We define \(\cal {F}\)  \( := \lbrace  G : K_{p, \, q} \subseteq G \subseteq K_p \vee (qK_1),\)  where \(q \ge p + 1 \ge 3\rbrace \) .
A cycle \(C\)  in a graph \(G\)  is called a Hamiltonian cycle of \(G\)  if \(C\)  contains all the vertices of \(G\) .
A graph \(G\)  is called Hamiltonian if \(G\)  has a Hamiltonian cycle.
